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Cascading Failures in Production Networks∗
David Rezza Baqaee†
August 24, 2015
Abstract
I show how the extensive margin of firm entry and exit can greatly amplify idiosyncraticshocks in an economy with a production network. I show that input-output models with entryand exit behave very differently to models without this margin. In particular, in such models,sales provide a very poor measure of the systemic importance of firms or industries. I derive anew notion of systemic influence called exit centrality that captures how exits in one industrywill affect equilibrium output. I show that exit centrality need not be monotonically relatedto an industry’s sales, size, or prices. Unlike the relevant notions of centrality in standardinput-output models, exit centrality depends on the industry’s role as both a supplier and as aconsumer of inputs. Furthermore, I show that granularities in systemically important industriescan cause one failure to snowball into a large-scale avalanche of failures. In this sense, shockscan be amplified as they travel through the network, whereas in standard input-output modelsthey cannot.
Keywords: Propagation, networks, macroeconomics, entry and exit
∗First Version: September 2013. I thank Emmanuel Farhi, Marc Melitz, Alp Simsek, and John Campbell for theirguidance. I thank Daron Acemoglu, Pol Antras, Natalie Bau, Aubrey Clark, Ben Golub, Gita Gopinath, Alireza Tahbaz-Salehi, Thomas Steinke, Oren Ziv, and David Yang, as well as many seminar participants, for helpful comments. I thankLudvig Sinander for his detailed comments on an earlier draft.
†London School of Economics and Political Science, [email protected].
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1 Introduction
In this paper, I investigate how the entry and exit of firms can act to amplify and propagate shocksin production networks. I model cascades of failures among firms linked through a productionnetwork, and show which firms and industries are systemically important for aggregate outcomes.I show that the extensive margin of entry is a powerful propagation mechanism of shocks throughproduction networks.
The idea that key consumers or key suppliers can have outsized effects on an economy if theyenter or exit has long been popular with policy makers and the public. Policies as diverse asagricultural subsidies, industrial policy, and government bail-outs frequently have been justifiedon the grounds of systemic importance. In academic economics, such ideas have been aroundsince at least the time of Hirschman (1958) and Rosenstein-Rodan (1943). However, despite theirpopularity with policy makers, these ideas have not been as influential in macroeconomics. Resultslike those of Hulten (1978) suggest that the systemic importance of firms and industries can beapproximated by their sales, even in the presence of linkages between firms, and therefore appealsto systemic importance are at best misguided and at worst dishonest.
However, in recent times, and motivated by current events, these ideas have experienced aresurgence in macroeconomics. After the failure of Lehman Brothers in the United States, theexistence of systemically important financial institutions, or SIFIs, became well-accepted in thefinance and macro-finance literatures.
Furthermore, recent work has also highlighted that systemic importance need not be limited tofinancial relationships. A memorable example of this is case of the US automobile manufacturingindustry, as discussed by Acemoglu et al. (2012). In the fall of 2008, the president of Ford MotorCo., Alan Mulally, requested government support for General Motors and Chrysler, but not forFord. The reason Mullaly wanted government support for his company’s rivals was that the failureof GM and Chrysler were predicted to result in the failure of many of their suppliers. Since Fordrelied on many of those same suppliers, Mulally feared that the knock-on effect on its supplierscould put Ford itself out of business. Mulally’s actions go against the standard expectation thatFord would benefit from the failure of its fiercest rivals, and the sales of GM did not tell us the fullstory about the potential consequences of GM’s failure.
While a government bailout of GM and Chrysler prevented a tide of failures in the UnitedStates,1 a scenario similar to the one Mulally described did play out in Australia. In May 2013,Ford Australia announced it they would stop manufacturing cars in 2017. Seven months later,GM Australia announced it would also stop manufacturing cars in 2017. Three months after that,in February 2014, Toyota Australia also announced that it would close its manufacturing plantsat the same time. This effectively ended automobile manufacturing in Australia. The Australiangovernment predicted that this would result in the loss of over 30,000 jobs, a figure they arrived
1See Goolsbee and Krueger (2015).
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at by adding the number of people directly employed by the three automakers and the Australiancar parts industry.
Empirical work by Carvalho et al. (2014) provides further support for the idea that firm entryand exit have important spill-overs on other firms through supply and demand chains. Carvalhoet al. (2014) instrument for firm exits using the 2011 Tohoku earthquake, and then show that firmexits have important negative spill-over effects on both upstream and downstream firms, evenwhen these firms are not directly connected to the exiting firm. They show that exits upstream ordownstream from a firm reduce the firm’s profits and make the firm more likely to exit.
Despite the importance of these spill-over effects, standard macroeconomic models, even oneswith production networks, do not allow for such forces. In this paper, I explicitly incorporatethe extensive margin of firm entry into an input-output model of production, and show that theextensive margin alters the quantitative and qualitative properties of the model.
This paper contributes to the growing literature on the microeconomic origins of aggregatefluctuations. Two recent strands of this literature are the granular origins hypothesis of Gabaix(2011) and the network origins hypothesis of Acemoglu et al. (2012). The former posits that shocksto large, or granular, firms can have non-trivial effects on real GDP. The latter posits that shocksto highly-connected firms can have non-trivial effects on real GDP. In practice, both theoriesemphasize a firm or industry’s size, as measured by its share of sales, as being the relevant measureof that player’s importance. The intuition here is that if a firm is systemically important, then inequilibrium, it will have a large fraction of total sales. However, as discussed above, we mightexpect interconnectedness to matter in ways that are not captured by prices or total quantities. Inthis paper, I show that allowing for entry and exit means that the network structure of productioncan amplify shocks and determine systemic importance in ways that are not captured by size.Furthermore, by combining granularity, networks, and the extensive margin, we can producedramatic domino chain reactions that would not be present in the absence of any one of these threeingredients.
To place my analysis in the broader literature, I first show that standard input-output macroe-conomic models that follow Long and Plosser (1983), like Acemoglu et al. (2012), Atalay (2013),or Baqaee (2015) have three crucial limitations. First, I show that these models have the propertythat their responses to idiosyncratic productivity shocks can be summarized in terms of exoge-nous sufficient statistics. Once we compute the relevant sufficient statistics, we can discard thenetwork structure. So, for example, I show that there are disconnected economies, with differenthouseholds tastes, that behave precisely like the network models. This result is similar in spirit tothe irrelevance result of Dupor (1999).
Second, I show that the relevant sufficient statistics are closely related to the equilibrium size offirms. This means that, without the extensive margin, it is not the interconnections per se, but howthose interconnections affect a firm’s size that determines a firm’s systemic importance. If we can
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arrive at the same sufficient statistics using a different (perhaps degenerate) network-structure,the equilibrium responses will be the same. This fact explains why the theoretical implicationsof the granular hypothesis of Gabaix (2011), where business cycles are driven by large firms, areobservationally equivalent to the theoretical implications of the network hypothesis of Acemogluet al. (2012), where business cycles are driven by well-connected firms.
Third, I show that in canonical input-output models, systemic importance depends only on afirm’s role as a supplier. In other words, as long as firms i and j have the same strength connectionsto the same customers, then their systemic influence will be the same, regardless of what i andj’s own supply chains look like. This contrasts with our starting anecdote about the Americanautomobile industry because it was not GM’s role as a supplier of cars, but its role as a consumer ofcar parts, that made GM systemically important. Canonical input-output models cannot be usedto understand episodes where a productivity shock travels from consumers upstream to suppliersand then travels back downstream to other consumers.
When we allow for entry and exit, all three of these limitations disappear. First, the sufficient-statistic approach breaks down. This is because the extensive margin makes “systemic importance”an endogenously determined value that is not well-approximated by equilibrium size or prices. Afirm that may seem like a small player, when measured by sales, can have potentially large impactson aggregate outcomes. On the other hand, a firm that may seem like a key player, as measuredby sales, can have relatively minor effects on the equilibrium. Furthermore, the endogenously-determined measures of systemic influence depend on a firm’s role as both a supplier and asa consumer, as well as on how many close substitutes there are for the firm. The fact thatsystemic importance is unrelated to sales contrasts sharply with the neoclassical world of Hulten’stheorem. Hulten (1978) shows that with arbitrary constant-returns production functions andarbitrary network structures, the effect on output of a marginal TFP shock to an industry dependsonly on that industry’s sales. This result is no longer true when we have a meaningful entry-exitmargin.
The model presented in this paper also allows us to combine the granular hypothesis of Gabaix(2011) and the network hypothesis of Acemoglu et al. (2012) in a new and interesting way. Inparticular, when firms are not infinitesimal, extensive margin shocks can be locally amplified viainterconnections. In other words, when a large and well-connected firm exits, it can set off anavalanche of firm failures that actually gets larger as it gathers steam. This type of amplification isimpossible in canonical models, where shocks can only decay as they travel through the network.
This paper also contributes to the wider literature on diffusion in network theory, by bridgingthe gap between two alternative modelling traditions. Loosely speaking, there are two popularapproaches to modelling diffusion on social networks. First, there are continuous models ofdiffusion like Katz (1953). Here, nodes influence each other in continuous ways – shocks travelaway from their source like waves and slowly die out. The strength of the connections between the
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nodes controls the rate of decay. Such shocks, sometimes called pulse processes, are characterizedby geometric sums. It is well-known that such models are incapable of local amplification: a shockto one node will always have its largest effect at its source, and the shock will decay as it travelsthrough the connections.2 These structures were first studied by Leontief (1936) in his input-outputmodel of the economy.
The other tradition consists of models that behave discontinuously, as typified by Morris (2000)or Elliott et al. (2012). In this class of models, sometimes called threshold models, each node has athreshold and is either active or inactive. When a node crosses its threshold it changes states and,by changing states, pushes its neighbors closer to their thresholds. Such models are frequentlyused to study the spread of epidemics, products, or even ideas. One of the earliest and mostinfluential threshold models is the Schelling (1971) model of segregation. Threshold models donot have wave-like properties since the rate at which shocks decay is not geometric. Crucially,these models are capable of generating local amplification – that is, shocks can be amplified asthey travel through the network. Unfortunately, these models are notoriously difficult to analyze.
In this paper, I consider a model that bridges the gap between the continuous and discretemodels. Specifically, I explicitly account for the mass of firms in a given industry. Industries witha continuum of firms behave continuously – the mass of firms responds continuously to shocks.On the other hand, lumpy industries, with only a few firms, behave discontinuously. For instance,a negative shock to an unconcentrated industry, say bakeries, will result in some fraction of bakersexiting. The fraction exiting will be a continuous function of the size of the shock. The effecton a neighboring upstream industry, like flour mills, or downstream industry, like cafes, will beattenuated by the strength of its connections to bakeries. However, a negative shock to a highlyconcentrated industry, like automobile manufacturing, will have no effect on the number of firmsunless it is large enough to force an exit. But once a large firm exits, it imparts an additionalimpulse to the size of the shock, which can trigger a cascade. Because the model is flexible enoughto express both behaviors, I can provide conditions under which we would expect a continuousapproximation to a discontinuous model to perform well.
The idea of cascading – domino-like – chain reactions also appears outside of economics. Inparticular, models of contagion and diffusion like the threshold models considered by Kempe et al.(2003) have these features. In these models, notions of connectedness play a key role since theonly way contagion can spread is via connections between nodes. An interesting implication ofembedding a contagion model into a general equilibrium economy is the role prices and aggregatedemand play – a role that does not have analogues in other threshold models. Typically, in athreshold model, shocks can only travel along edges. Contagion can only spread to nodes whoare connected to an infected node. This important intuition breaks down in general equilibriummodels since all firms are linked together via aggregate supply and demand. This means that
2See, for example, page 253 in Berman and Plemmons (1979) and also appendix D.
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general equilibrium forces can act like long-distance carriers of disease. Shocks in one fragileindustry, like the financial industry, can jump via aggregate demand or supply, to a differentfragile industry like automobile manufacturing, even if these two industries are not connected.
This paper is belongs to the literature seeking to derive aggregate fluctuations by combiningmicroeconomic shocks with local interactions. Empirical work on this front includes Foersteret al. (2011), Di Giovanni et al. (2014), Barrot and Sauvagnat (2015), Carvalho et al. (2014), andAcemoglu et al. (2015). Theoretical work can be roughly divided into two categories. The firstcategory are papers that build on the multi-sector neoclassical model of Long and Plosser (1983),like Horvath (2000), Acemoglu et al. (2012), Atalay (2013), and Jones (2011). These are log-linearmodels of propagation, and follow the tradition of Leontief (1936) in having linear-geometricdiffusion. As shown by Hulten (1978), the propagation of TFP shocks in such models is, atleast locally, tied to the distribution of firm sizes. This means that in order for a firm to besystemically important, it must have high sales. The second category are models that do notsatisfy the requirements of Hulten’s theorem. These models, like Jovanovic (1987), Durlauf (1993),and Scheinkman and Woodford (1994), feature strong nonlinearities, their sales distribution is nota decisive sufficient statistic, and they do not feature the kind of geometric patterns of diffusionfound in input-output models. However, these models depart dramatically from the standardassumptions of macroeconomics and can be difficult to work with. This paper has elements incommon with both of these literatures. The individual ingredients in my model are very familiar tomacroeconomists – monopolistic competition, free-entry, and input-output linkages via constant-elasticitity-of-substitution production functions. However, the presence of all three ingredientstogether results in strongly nonlinear interactions and an uncoupling of systemic importance fromthe distribution of sales.
The structure of paper is as follows. In section 2, I set up the model and define its equilibrium.In section 3, I characterize the equilibrium conditional on the mass of entrants in each industryand define some key centrality measures. In section 4, I study how the model behaves when theextensive margin of firm entry and exit is shut down. I prove results showing that the networkstructure can be summarized by sufficient statistics related to size. I also show that we can thinkof these models as non-interconnected models with different parameters. In section 5, I allowfor firm entry and exit. First, I characterize the model’s responses to shocks in the limit whereall firms are massless. I show that sufficient statistics are no longer available, and I derive anendogenous measure of influence called exit centrality. Then, I consider the case when firms canhave positive mass, and show that with atomistic firms, shocks can be locally amplified. I provean approximability result showing conditions under which we can approximate an intractablediscontinuous model using a tractable continuous model. Motivated by Ford’s request for abailout of GM, I also consider whether efficient bail-outs can be identified by surveying firms.Specifically, when can we trust one firm’s testimony about whether or not another firm should be
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bailed out. I conclude in section 6.
2 Model
In this section, I spell out the structure of the model and define the equilibrium. There are threetypes of agents: households, firms, and a government. Each firm belongs to an industry, andthere are N industries. There are two periods: entry decisions happen simultaneously in period 1,production and consumption take place in period 2.
Household’s Problem
The households in the model are homogenous with a unit mass. The representative householdmaximizes utility
U(c1, . . . , cN) =
N∑k=1
β1σ
k cσ−1σ
k
σσ−1
,
where ck represents composite consumption of varieties from industry k and σ > 0 is the elasticityof substitution across industries. The weights βk ≥ 0 determine household tastes for goods andservices from the different industries. The composite consumption good produced by industry kis given by
ck =
M−ϕkk
Nk∑i=1
∆kc(k, i)εk−1εk
εkεk−1
,
where c(k, i) is household consumption from firm i in industry k and εk > 1 is the elasticity ofsubstitution across firms within industry k. Here, Nk is the number of firms active in industry kand ∆k is the exogenous weight of each firm.3 The total mass of varieties in industry k is givenby Mk = Nk∆k. Finally, the term M−ϕk
k controls the strength of the love-of-variety effect. If we letϕk = 1/εk, then there is no love-of-varieties, and if we let ϕk = 0, we get the standard Dixit andStiglitz (1977) demand system.4 The household’s budget is given by∑
k,i
p(k, i)c(k, i)∆k = wl +∑k,i
π(k, i)∆k − τ,
where p(k, i) is the price of firm i in industry k and π(k, i) is firm i in industry k’s profits. The wageis w and labor is inelastically supplied at l. For the rest of the paper, and without loss of generality,
3If we take the limit ∆k → 0, this sum converges to a Riemann integral.4See section 5.1 for more discussion about the significance of the love-of-varieties effect in this model. Love of
varieties can be thought of as a reduced form for a more complex model where increasing product variety improvesthe match between products and consumers. Anderson et al. (1992) provide a microfoundation for this via a discretechoice model where each consumer consumes only a single variety, but there exists a CES representative consumer forthe population of consumers.
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we take labor to be the numeraire so that w = 1, and fix the supply of labor l = 1. Lump sum taxesby the government are denoted τ.
Firms’ Problem
Let firm i in industry k have profits
π(k, i) = p(k, i)y(k, i) −N∑
l=1
Nl∑j
p(l, j)x(k, i, l, j)∆l − wh(k, i) − w fk + τk,
where p(k, i) is the price and y(k, i) is the output of the firm. Inputs from firm j in industry l arex(k, i, l, j) and labor inputs are h(k, i). Finally, in order to operate, each firm must pay a fixed cost fkin units of labor and the firm potentially receives a lump sum subsidy τk. The firm’s productionfunction (once the fixed cost has been paid) is constant-returns-to-scale
y(k, i) =
α 1σ
k (zkl(k, i))σ−1σ +
N∑l=1
ω1σ
k,lx(k, i, l)σ−1σ
σσ−1
.
Here, σ > 0 is again the elasticity of substitution among inputs, and ωkl is the share parameter forhow intensively firms in industry k use composite inputs from industry l. The N×N matrix of ωkl,denoted by Ω, determines the network-structure of this economy. One can think of this matrix asthe adjacency matrix of a weighted directed graph. The parameter αk > 0 gives the intensity withwhich firms in industry k use labor (net of fixed costs).
Labor productivity shocks, like the ones considered by Acemoglu et al. (2012) and Atalay (2013)are denoted by zk. Note that when σ , 1, a productivity shock zk to industry k is equivalent tochanging that industry’s labor intensity from αk to αkzσ−1
k . Therefore, as long as σ , 1, we canthink of αk as including both the productivity shock and the labor intensity. This way we do notneed to directly make reference to the shocks z since they are just equivalent to changing α. Thisequivalence breaks down when σ = 1, and in those cases, we shall have to work directly with z.For the majority of this paper, I focus on the propagation of these productivity shocks. The samemethods can easily be used to study other shocks like fixed-cost shocks or demand shocks.
The composite intermediate input from industry l used by firm i in industry k is
x(k, i, l) =
M−ϕll
Nl∑j=1
∆lx(k, i, l, j)εl−1εl
εlεl−1
,
where εl is the elasticity of substitution across different firms within industry l. As before, the termM−ϕl
l affects the strength of the love-of-varieties effect in industry l. Letting ϕl = 0 gives us thetraditional Dixit-Stiglitz formulation. Note that the elasticities of substitution are the same for all
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users of an industry’s output.
Government’s Problem
The government runs a balanced budget so that∑k
τkMk = τ.
For now, we assume that the government is completely passive. In section 5.5, we return to howthe government can vary its taxes and subsidies to affect the equilibrium.
Notation
Let ei denote the ith standard basis vector. Let Ω be the N×N matrix whose i jth element is equal toωi j. Let α and β be the N × 1 vectors consisting of αis and βis. Let M be the N ×N diagonal matrixwhose ith element is Mi, and let M be the N × N diagonal matrix whose ith diagonal element is
equal to M1−ϕiεiεi−1
i . Let denote the element-wise or Hadamard product, and diag : RN→ RN2
be theoperator that maps a vector to a diagonal matrix. To simplify the notation, when a variable appearswithout a subscript, this represents the matrix of all such variables, and the object vs, where v is avector and s is a scalar, should be interpreted as the vector v raised to the power of s elementwise.
Definition 2.1. An economy E is defined by the tuple E = (β,Ω, α, ε, σ, f ,∆, ϕ). The vector βcontains household taste parameters, Ω captures the input-output share parameters, α containsthe industrial labor share parameters, ε > 1 is the vector of industrial elasticities of substitution,σ > 0 is the cross-industry elasticity of substitution, f is the vector of fixed costs, ∆ is the vector ofmasses of firms in each industry, and ϕ is the parameter controlling the love-of-variety effect.
Equilibrium
We study the subgame perfect Nash equilibrium. In period 1, potential entrants make simultaneousentry decisions. In period 2, firms play general equilibrium conditional on period 1’s entrydecisions. Define a markup function µ : RN
× RN→ RN to be a function that maps number of
entrants Nkk and productivity shocks zk to markups for each industry. Now we can define generalequilibrium as follows. I assume that firms set their prices to equal the markup function timestheir marginal cost.
Definition 2.2. A general equilibrium of economy E is a collection of prices p(i, k), wage w, andinput demands x(i, k, l, j), outputs y(i, k), consumptions c(i, k) and labor demands l(i, k) such thatfor number of entrants Nk
Nk=1, vector of productivity shocks z, and markup function µ(N, z):
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(i) Each firm minimizes its costs subject to demand for its goods given the markup function,
(ii) the representative household chooses consumption to maximize utility,
(iii) the government runs a balanced budget,
(iv) markets for each good and labor clear.
Note that prices and quantities in a general equilibrium depend on the number of entrants Nk,the vector of productivity shocks, and the assumed industry-level markups µ. Conditions (ii)–(iv)are standard, but condition (i) merits further discussion. Condition (i) is deliberately vague. Itsimply requires that the firms minimize their costs, given the demand they face and leaves theirchoice over prices unspecified – it simply requires that there be some function µ that determinestheir markups as a function of the number of entrants Nk and the productivities z.
The exact nature of the markup function µ, which will depend on what we assume aboutcompetition in each industry, will play an important role in our analysis. However, for now, wecan leave the market structure general. Note that by choosing µ correctly, we can represent manydifferent popular models of competition including Cournot, Dixit-Stiglitz, and perfect competition.
Let Π : RN+ × RN
+ → RN be the function mapping the masses of entrants M and vector ofproductivity shocks z to industrial profits assuming general equilibrium in period 2. In theorem3.5, I analytically characterize this function.
Definition 2.3. A vector of integers NkNk=1 is an equilibrium number of entrants if
Πi(Mi,M−i, z) ≥ 0 > Πi(Mi + ∆i,M−i, z) (i ∈ 1, 2, . . . ,N),
where M j = N j∆ j, for all j.
Intuitively, a vector of integers is an equilibrium number of entrants if all firms make non-negative profits, and the entry of an additional firm in any industry results in firms in that industrymaking negative profits. Note that if we let ∆i → 0, the entry inequalities become a zero-profitequality condition.
Before analyzing the model, it helps to define some key statistics. These are standard definitionsfrom the literature on monopolistic competition. See, for example, Bettendorf and Heijdra (2003).
Definition 2.4. The price index for industry k is given by
pk =
M−ϕkεkk
Nk∑i
∆kp(k, i)1−εk
1
1−εk
,
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and the total composite output of industry k is given by
yk =
M−ϕkk
Nk∑i
∆ky(k, i)εk−1εk
εkεk−1
.
The consumer price index, which represents the price level for the household, is given by
Pc =
N∑k
βkp1−σk
1
1−σ
,
and total consumption by the household is given by
C =
N∑k=1
β1σ
k cσ−1σ
k
σσ−1
,
which is just the utility of the household.
These are the “ideal” price and quantity averages for each industry. The reason we do notsimply average prices to get a price index or add outputs to get total output is because evenwithin each industry, each firm is producing a slightly different product. When the elasticity ofsubstitution εk = 0, then the price index for that industry is simply the sum of all the prices,since there is no substitution and a consumer of this industry must buy all the varieties. Whenthe elasticity of substitution εk → ∞, the price index for an industry is the minimum price, sincehouseholds will only purchase from the cheapest firm in each industry. An important special caseis when εk → 1, where the price index is a geometric average of the industrial prices.
3 General Equilibrium Subgame
In this section, I characterize the equilibrium in the general equilibrium subgame of period 2,conditional on the number of entrants in each industry. I introduce two related notions of centrality,the supplier and consumer centrality, and I show that together, they determine the size distributionof industries in the economy.
Let us define supply-side centrality measure.
Definition 3.1. The supplier centrality is
β′ = β′Ψs = β′∞∑
n=0
(Mσ−1µ−σΩ
)n, (1)
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where
Ψs = (I − Mσ−1µ−σΩ)−1 =
∞∑n=0
(Mσ−1µ−σΩ
)n,
is the supply-side influence matrix.
We can think of β as the network-adjusted consumption share of the industries. The kth elementof β captures demand from the household that reaches industry k, whether directly or indirectlythrough other industries who use k’s products. Each term of the sum in (1) is interpretable. Thefirst term captures the direct intensity with which the household buys from an industry, the secondterm captures the indirect effect when the household consumes something that used the industryas an input, and so on. In other words, supplier centrality captures the importance of the industryas a supplier to the household. The following helps establish why we might care about β
Lemma 3.1. In equilibrium,
βi =
(pσi yi
Pσc C
),
where Pc is the ideal price index for the household, C is total consumption by the household, pi is industryi’s ideal price index and yi is industry i’s composite output.
Note that in the Cobb-Douglas limit, where the elasticity of substitution across industries σ isequal to one, βi is precisely an industry’s share of sales. In the Cobb-Douglas case, β coincides withthe influence measure defined by Acemoglu et al. (2012).
There are two reasons to think of β as a supplier centrality. First, since it measures the intensitywith which the household consumes goods from an industry (both directly and indirectly throughintermediaries), β is high the more frequently an industry appears in the households’ supplychains. So, all else equal, star suppliers, who supply many firms, have higher β. Secondly, thevalue of βk depends only on who the industry k sells to, and not on who it buys from.
Proposition 3.2. Consider two industries k and l such that
ω jk = ω jl, ( j = 1, . . . ,N), βk = βl, µk = µl, Mk = Ml,
then βk = βl. In other words, if two industries have the same immediate customer base, their supplier-centralities are the same.
Note that with entry and exit,
β′ = β′(I − Mσ−1µ−σΩ)−1,
is endogenous. In particular, a change in the mass of firms affects both the love-of-variety effectM and potentially the markups µ. These in turn change the value of β in highly nonlinear ways.
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Intuitively, to get interesting results, entry and exit have to affect the strength of connectionsbetween industries. The centrality of industries in the network must change as firms enter and exitthe market. The connections encapsulated in Ω have to become stronger or weaker depending onthe number of entrants in each industry. If we make expenditure shares exogenous, by shuttingdown love-of-varieties and imposing constant markups, then β will not respond to extensivemargin shocks.
In section 4, I show that when the extensive margin is shutdown, β captures the response ofoutput to productivity shocks. This is a generalization of a result in Acemoglu et al. (2012), whoshow that in a Cobb-Douglas model without an extensive margin, β captures the extent to whichindustry-specific labor productivity shocks affect output. The intuition for this result is clear: if afirm is supplying a large fraction of the economy, then its productivity shocks have a large impacton output.
Now let us define the analogous demand-side consumer centrality.
Definition 3.2. The consumer centrality is
α = Ψd
(α zσ−1
),
where
Ψd = (I − µ1−σMσ−1Ω)−1µ1−σMσ−1 =
∞∑n=0
(µ1−σMσ−1Ω
)nµ1−σMσ−1,
is the demand-side influence matrix.
This is the flip-side to the supply-side centrality measure. It captures how frequently anindustry appears downstream from the factors of production (in this case labor).5 Whereas βk
depended only on who bought from k, the consumer centrality αk depends only on who k buysfrom.
Proposition 3.3. Consider two industries k and l such that
ωkj = ωl j, ( j = 1, . . . ,N), αk = αl, µk = µl,Mk = Ml,
then αk = αl. In other words, if two industries have the same immediate supplier base, their consumercentralities are the same.
As with the supplier centrality, note that with entry and exit,
α = (I − µ1−σMσ−1Ω)−1µ1−σMσ−1α,
5As discussed before, to simplify notation, we can simply subsume the effect of z into α, writing only α = Ψdα.
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is endogenous. In particular, a change in the mass of firms affects both the love-of-variety effectM and potentially the markups µ. These in turn change the value of α in highly nonlinear ways.As with the supplier centrality, if we make expenditure shares exogenous, by shutting downlove-of-varieties and imposing constant markups, then consumer centrality α does not respond toextensive margin shocks.
Consumer centrality α is closely related to the network-adjusted labor intensity measure de-fined by Baqaee (2015). One can show that in a model without an extensive margin, α capturesthe response of output to demand shocks.6 The following lemma shows why we might care aboutconsumer centrality α.
Lemma 3.4. In equilibrium,
αi =(pi
w
)1−σ,
where pi is industry i’s price index and w is the nominal wage.
The intuition is that αk captures all the ways industry k uses labor, both directly, and indirectlythrough its suppliers, suppliers’ suppliers, and so on. In other words, we can think of αk asthe network-adjusted factor costs of k. The higher αk is, the longer, more competitive, and morevalue-intensive the supply chains of k are, and therefore, the lower its marginal costs and prices.This is why αk depends on industry k’s supply-chain, since it is suppliers and not consumers, whocontribute to marginal costs.
A key result of this paper, and one that delivers much of the intuition for the results, is thefollowing characterization of active firms’ profit functions in terms of supplier and consumercentrality measures:
Theorem 3.5. The profit of firm i in industry k are equal to
π(k, i) =µk − 1µk
1Mk︸ ︷︷ ︸
industrial competition
× PcC︸︷︷︸aggregate demand
×
( wPc
)1−σ
︸ ︷︷ ︸aggregate supply
× βk︸︷︷︸supplier centrality
× αk︸︷︷︸consumer centrality
− w fk︸︷︷︸fixed cost
.
As promised, Mkπ(k, i) analytically characterizes industry k’s profits Πk.The expression in theorem 3.5 tells us that the profits of a firm are determined by a few intuitive
key statistics. The product of βk and αk, which are the supply-side and demand-side centrality ofthe industry, give us an industry’s share of sales. The nominal GDP term PcC is an economy-wideshifter of all industry’s profits, akin to aggregate demand. The terms µk−1
µkand Mk convert an
industry’s sales into firm-level gross profit. Finally, we arrive at a firm’s net profits by subtractingthe fixed costs of entry from its gross profit.
6As long as the factor of production is not inelastically supplied.
14
Path Example
Before moving on to an analysis of the equilibrium, first let us demonstrate the intuition so farusing a simple example of a production chain, shown in figure 1.
HH
1 2 · · · N
l1 + π1
l2 + π2
li + πi
lN + πN
Figure 1: A production path example. The solid arrows represent the flow of goods and services,and the dashed arrows indicate the flow of money. The household HH buys from industry 1 whobuys from industry 2 and so on. Each industry in turn pays labor income and rebates profits tothe household.
Begin by computing the supplier-centrality for industry k in this chain:
βk = β′(I − Mσ−1µ−σΩ)−1ek,
=
k−1∏i=0
(1 − αi)Mσ−1i µ−σi .
Note that, for this example, βk is a product. Therefore, if any downstream industry i < k disappears,so that Mi = 0, then the supplier centrality of industry k drops to zero. This is intuitive, since ifa downstream industry collapses, that cuts all upstream industries off from any demand. Thecentrality of k as a supplier is increasing in the strength of its downstream connections (1 − α) anddecreasing in the size of downstream markups µi. The latter represents double-marginalizationin this economy. Note that as long as the elasticity of substitution is greater than 1, an industry’ssupplier centrality is increasing in the mass of downstream industries. Intuitively, when theelasticity of substitution is greater than one, more industries downstream attract more demandfrom the household. Lastly, observe that, as implied by proposition 3.2, the kth industry’s suppliercentrality depends purely on its customers, customers’ customers, and so on. It does not dependon its suppliers.
Now, compute the consumer-centrality for node k in this chain:
αk = e′k(I − Mσ−1µ1−σΩ)−1Mσ−1µ1−σα,
=
N∑j=k+1
j−1∏i=k
(1 − αi)Mσ−1i µ1−σ
i
α j + αkMσ−1k µ1−σ
k .
15
The consumer centrality is slightly more complex. It is an arithmetico-geometric. The intuitionis that even if an upstream supplier i for industry k disappears, industry k still has access to allsuppliers j where j < i. But any supplier j > i also drops out. This gives rise to the arithmetic serieswhere each term is a product. Once again, the consumer centrality is increasing in the strength ofthe connection. As long as the elasticity of substitution is greater than one, consumer centrality isdecreasing in markups and increasing in the mass of upstream firms.
When the elasticity of substitution is less than one, consumer centrality is increasing in upstreammarkups. This sounds perverse until we recall that proposition 3.1 implies that
αk =(pk
w
)1−σ.
Therefore, when the elasticity of substitution is less than one, higher consumer centralities indicatehigher prices, not lower prices. Therefore, it is intuitive that in this case, higher upstream markupscorrespond to higher consumer centrality, since this corresponds to higher prices. Finally, notethat, as implied by proposition 3.3, consumer centrality of a firm depends only on who it buysfrom, and not on who it sells to.
4 Equilibrium without the Extensive Margin
Before moving onto the case with the extensive margin, let us consider the behavior of this modelwhen the extensive margin is shut down. This makes the model into a generalization of thecanonical input-output model of Long and Plosser (1983). I show that in this class of models, anindustry’s influence on other industries and on the aggregate economy is exogenously determined,and that it is only dependent on how important the industry is as a supplier of inputs. Furthermore,I show that a firm’s size is an important determinant of a firm’s influence.
Standard models that lack an entry margin are typically perfectly competitive, with no fixedcosts, so that the representative firm in each industry makes zero profits. To get to this benchmark,let fi = 0 for every industry so that there are no fixed costs, and let εi → ∞ so that all industriesare perfectly competitive and firms have no market power. Then, due to constant returns to scale,the size of firms in each industry is indeterminate. Without loss of generality, we can fix M = Iso that there is a unit mass of firms in each industry. Technically, there may be entry or exit,but since firms in each industry are completely homogenous, entry and exit are observationallyequivalent to firms getting larger or smaller. That is, the extensive and intensive margins areindistinguishable. Lastly, given the lack of product differentiation and free entry, there are nomarkups in this economy µ = I. Note that in this special case, the supply-side and demand-sideinfluence matrices are equal Ψd = Ψs = (I −Ω)−1.
Let us consider real GDP C(z|E) as a function of productivity shocks z for an economy E.
16
Theorem 4.1. For every household share parameter vector β and input-output share parameter matrixΩ, the non-interconnected economy with household share parameter vector β and degenerate input-outputshare parameter matrix 0, we have
C(z|β,Ω) = C(z|β, 0).
This result means that the existence of input-output connections is isomorphic to a change inhousehold share parameters. This underlies the earlier claim that, it is not the interconnectionsthemselves, but how intensively the household ultimately consumes goods from each industrythat determines the model’s equilibrium responses to shocks. Furthermore, by lemma 3.1, βcorresponds to a measure of industry sizes in equilibrium.
Proposition 4.2. Around the steady-state, where zk = 1 for all k, we have that
βk
βl=
pkyk
plyl,
so although β is not equal to share of sales everywhere, at the steady-state, it does reflect an industry’s sales.
Even though β no longer corresponds to an industry’s size globally, it is still exogenous, anddepends solely on industry’s role as a supplier. Furthermore, the exact nature of the network-structure is irrelevant since an industry i could be big because it sells a lot to the household (largeβi) or because it supplies many other firms (the ith column of Ψs is large).
The fact that I assumed perfect competition and zero fixed costs are not necessary for thequalitative nature of the results in this section, they simply make the results much cleaner. InAppendix A, I show how allowing for fixed costs and monopolistic competition, but not allowingentry, would affect the results of this section.
5 Equilibrium with Extensive Margin
Now that we understand the properties of the model without the extensive margin, let us considerhow entry and exit changes the model’s properties. In this case, the network structure is endoge-nous since the number of firms in each industry is endogenous. This means that our notions ofcentrality β and α are determined in equilibrium, and they change in response to shocks. It will nolonger be the case that an industry’s influence is exogenous, nor will its influence depend solelyon its role as a supplier. With entry, an industry’s influence depends both on its in-degrees (arrowspointing in) and its out-degrees (arrows pointing out), and these are determined in equilibriumdepending on the mass of firms in other industries. Two firms with identical roles as suppliers canhave markedly different effects on the equilibrium depending on their roles as consumers.
Our analysis will proceed in steps. First, I discuss the market structure and the mechanismsthrough which entry can affect the equilibrium. Then, in section 5.2, I consider the limiting case
17
of the model where all firms have zero mass. This means that the mass of firms in each industrywill continuously adjust to ensure that all active firms make zero profits in equilibrium. This willallow for sharp analytical results about the equilibrium response to marginal shocks. Once wehave characterized the properties of the continuous limit, we consider the case where firms canhave positive mass ∆ > 0 in section 5.3. In this case, the model inherits some of the cascadingand amplification properties of linear threshold models like Schelling (1971), but it also retainsthe linear geometric structure of Leontief (1936). Most of the time, the model can be expected tobehave like its continuous limit; however, occasionally, this approximation can dramatically breakdown.
The intuition is that each industry can support an integer number of firms. If firms aresufficiently small and there are many of them in an industry, then a shock to the industry willcontinuously perturb the mass of firms in that industry. However, if there are few firms in anindustry, then certain shocks can cause a large firm to discontinuously exit or enter. This adds anadditional impulse to the original shock, which will now travel to the neighbors of the affectedindustry with more force. This process can feed on itself as a small firm’s failure can trigger a chainreaction that results in a large mass of firms exiting.
A full characterization of the equilibrium of the model with lumpy firms is not possible. Anatural solution is to use the continuous model to approximate the behavior of the model withlumpy firms. I show conditions under which the continuous limit provides a bad approximationto the discontinuous model. In particular, I show that the network can make firm’s payoffs non-monotonic in each other’s entry decisions. So, there are regions where entry decisions are strategiccomplements and and regions where they are substitutes. I show that when a shock moves firmsclose to this intermediate region, the approximation error will explode.
To foreshadow the formal model, consider an extremely stylized example in figure 2. Thehousehold HH is served by three industries: GM, F, and the unspecified rest of the economy. GMand F share a common supplier D. To make the intuition transparent, suppose that each industryconsists of a single firm. Recall that theorem 3.5 implies that in order for a firm to remain profitable,its gross profits
µk − 1µk
βiαiPσc C
have to be greater than an exogenous threshold fi. For this example, suppose that markups areconstant. Each term of β, α, and Pσc C can respond to a shock. Therefore, the shock to a firm cantravel via network connections β and α or it can travel through household demand Pσc C.
The example in figure 2 illustrates the various ways shocks can propagate. In panel 2a, I showa negative shock to the rest of the economy that reduces the aggregate demand term Pσc C. Then,because of this change in Pσc C, it can be the case that GM is forced to exit, and this is shown in panel2b. GM exiting will cause D’s supplier centrality βD to fall, and so D can be forced out, shownin panel 2c. Finally, once D is gone, this can cause F’s consumer centrality α to drop, which can
18
cause F to also exit. Of course, this is not a realistic calibration of the model, but it does illustratethe forces operating in the model. In a canonical input-output model, like the one in section 4, achange to Pσc C would have no network effects since β and α would be exogenous.
This example shows that the model is capable of formalizing the intuition for why a bail-outof GM may have been crucial. As pointed out by Goolsbee and Krueger (2015), in this scenario,it might be “essential to rescue GM to prevent an uncontrolled bankruptcy and the failure ofcountless suppliers, with potentially systemic effects that could sink the entire auto industry.”
HH
GM F
D
rest of economy ↓
(a) Shock to the rest of the economy occurs, and Pσc Cfalls.
HH
GM F
D
rest of economy ↓
(b) Effect on GM due to a fall in Pσc C.
HH
GM F
D
rest of economy ↓
(c) Effect on D through a fall in β due to GM’s exit.
HH
GM F
D
rest of economy ↓
(d) Effect on F through a fall in α due to D’s exit.
Figure 2: A toy representation of an economy where the direction of arrows denote the flowof goods. The node HH denotes the household. This shows how the shock can travel in bothdirections on the extensive margin.
5.1 Market Structure, Love of Varieties, and the Role of Markups
So far, we have not explicitly defined the structure of each market. Instead, by working with themarkups directly, we have avoided the need to specify the nature of competition. In fact, for manyof the results that are to follow, all we require is that the markups µk(N) in industry k depend onlyon the number of entrants in industry k. This assumption is enough to deliver all the qualitativeresults we need.
The novel mechanism in this model is that entry and exit affect the strength of links betweenindustries – this in turn affects the shape of the network, which plays itself out via the highlynonlinear changes in the supply-side and demand-side centrality measures. The reason entry and
19
exit affect the strength of connections in this paper can either be due to the “love of varieties”effect, where more product differentiation increases demand for goods from an industry, or dueto markups, where increased entry reduces markups. The key is that industry-level output andprices must respond to the number of entrants – as long as more entrants cause the industry’s priceand output to change, very similar forces will operate, regardless of the specific reason.
To see this, note that the influence matrices Ψs and Ψd, which capture the strength of theconnections are of the form
Ψs = (I − SΩ)−1 and Ψd = (I −DΩ)−1D,
where S and D are diagonal matrices. Specifically, S and D are products of the diagonal matrixconsisting of the mass effect M and the diagonal matrix of markups µ.
However, qualitatively, what matters is that the diagonal matrices D and S must respond tochanges in the mass of players. For instance, if increased entry into a market reduces markups,then the qualitative impact of this would be very similar to an increase in product variety. To seethis, we can remove the love of varieties effect by setting ϕk = 1/εk for every industry. This meansthat the love of variety term M drops out of the definitions, and the supplier centrality measure βbecomes
β′ = β′(I − µ−σΩ)−1,
and the consumer centrality measure α becomes
α = (I − µ1−σΩ)−1µ1−σα,
where the term M is now absent. However, as long as markups µ still respond to entry and exit,we will still get our desired effects.
As examples, let us work with three alternative market structures.
(a) Dixit-Stiglitz: Each firm chooses its price to maximize profit, taking as given the industrialprice level and industrial demand;
(b) Oligopolistic Competition: Each firm chooses its price to maximize its profit, taking as given theeconomy-wide price levels and demand;
(c) Cournot in quantities: Each firm chooses its quantity to maximize its profit, taking as given theeconomy-wide price levels and demand, and there is no product differentiation.
Condition (a) is the monopolistic competition assumption: firms ignore the impact of their behavioron the industry-level price level. This assumption is fully rational when each firm in the industryhas zero mass (the limiting case of ∆k → 0). Since in that case, an individual firm’s price has noeffect on the industry’s price level. However, when ∆k > 0, particularly, when ∆k is large relative
20
to Mk, then this assumption is less tenable. In that case, conditions (b) and (c), are more palatable.All these market structures result in markups that depend only on the number of other firms inthat industry. The choice between (a), (b), or (c) does not qualitatively change the nature of theresults, though it does matter quantitatively.
Proposition 5.1. Under condition (a), the markup function µ is given by
µk =εk
εk − 1.
Under condition (b), the markup function µ is given by
µk =εkNk − (εk − σ)
(εk − 1)Nk − (εk − σ).
Under condition (c), the markup function µ is given by
µk =σNk
σNk − 1.
We can also mix and match our market structure, assuming different market structures indifferent industries. As expected, the markups under (b) converge to those under (a) whenNk → ∞, since the price impact of each firm becomes negligible. Also observe that the markupsunder (c) converge to those under (a) when Nk →∞, since the firms are perfect competitors. Finally,note that markups under (b) and (c) are the same when Nk = 1, since the firm is a monopolist. Thequalitative nature of the results is not sensitive to the choice of the markup function µ, as long asµk depends only on Nk, and is decreasing in Nk.
5.2 Massless Limit
To begin with, let us first consider the equilibrium of the model in the limit where firm sizes ∆→ 0.This corresponds to the case where the mass of firms Mk in industry k adjusts so that firms inindustry k make exactly zero profits. This can be seen by taking the limit of the expression indefinition 2.3 as ∆ → 0. The primary motivation for looking at this limit is analytical tractability.By considering massless firms, we can glean useful intuition about the marginal effects of shockson the equilibrium.
For simplicity, I assume the Dixit-Stiglitz market structure with ψk = 0, so that markups areconstant and all the effects come through the love of product-variety. As stated before, this doesnot color the qualitative nature of the results. In appendix B, I show what the formulas would looklike for the general case.7
7The advantage of using love-of-variety is that the effects do not wash out when we take the limit to infinitesimalfirms, whereas the markup variation under Cournot and Oligopolistic competition disappear when Nk →∞. A simple
21
5.2.1 Out-of-Equilibrium Effect
To get intuition for the model’s properties, let’s consider the following out-of-equilibrium com-parative statics: how does industry k’s supplier centrality change when there is entry in industryi , k? This comparative static holds fixed the mass of firms in all industries except industry i.In equilibrium, of course, the masses in other industries would respond to the increased entry,but it helps our understanding if we first isolate the partial equilibrium effect. If we had lags inentry and exit, these partial equilibrium results would be relevant for understanding short-runeffects. Once we have characterized the out-of-equilibrium effects, we turn our attention to theequilibrium responses of the model.
Lemma 5.2. The derivative of βk with respect to a percentage change in the mass of firms in industry i,holding fixed the mass of firms in all other industries is given by
∂βk
∂ log Mi=
(σ − 1εi − 1
)βi(Ψs
ik − 1(i = k)).
This expression is very intuitive. The impact to industry k’s centrality as a supplier depends oni’s importance as a supplier β, and on how much i buys from k (whether directly or indirectly) Ψs
ik.So big effects are felt if i is a key supplier and i is also a key consumer for k. For ease of notation,denote the matrix of these partial derivatives by
∂β
∂ log(M1)= Ψ1.
Similar results apply to the consumer centrality measure.
Lemma 5.3. The derivative of αk with respect to a percentage change in the mass of firms in industry i,holding fixed the mass of firms in all other industries is given by
Mσi
∂αk
∂ log Mi=
(σ − 1εi − 1
) (εi
εi − 1
)σ−1αiΨ
dki.
This expression is the demand-side analogue to lemma 5.2. Recall that α is a measure of prices.Lemma 5.3 shows that the impact to industry k’s prices as depends on i’s price αi, and on howmuch i sells to k (whether directly or indirectly) Ψd
ki. So big effects are felt if i is a key consumerof inputs and i is a key supplier of inputs to k. Once again, the impact to industry k’s consumercentrality depends on industry i’s suppliers and i’s customers. For ease of notation, denote the
way to model variable markups with infinite firms would be to use the approach of Zhelobodko et al. (2012). They letthe elasticity of substitution εk depend directly on the mass of firms Mk. If we let εk be an increasing function of Mk,so that markups decline with more product-variety, then we can easily extend the results to the case where there areinfinitesimal firms, there is no love of variety, and the extensive margin matters due to its impact on markups instead.
22
matrix of partial derivatives by∂α
∂ log(M1)= Ψ2.
5.2.2 Path Example
Before moving on to an analysis of the general equilibrium, let us first demonstrate the intuition ofour partial equilibrium results using a simple example of a production chain depicted in figure 3.
1 2 . . . N
HH
Figure 3: The arrows represent the flow of goods and services. Let ωk−1,k be constant for all k, andβk, αk and εk be constant for all k.
In this example, each industry k sells some goods directly to the household, and some goods tothe industry below it k−1. For ease of exposition, consider β and α at the point where all industrieshave a unit mass of firms. In this example, βk is increasing in k and αk is decreasing in k. Thatis, industry N is the most central supplier and least central consumer, and industry 1 is the mostcentral consumer and least central supplier.
Now consider changing the mass of firms in industry N. Note that although industry N isthe most central supplier, so βN > βk for k , N, it has no effect on the supplier centrality of otherindustries
∂βk
∂MN= 0.
This is because industry N buys from no other industries. Therefore, despite having the mostdemand, its direct impact on how much demand reaches other industries is zero.
Now consider changing the mass of firms in industry 1. Industry 1 is the most central consumerso α1 > αk for k , 1. However,
∂αk
∂M1= 0 (k > 1),
because industry 1 sells to no other industries. Therefore, despite having the longest supply chain,its direct impact on the supply chains of other industries is zero.
This simple example illustrates why both the in-degrees and the out-degrees will matter forhow the centrality measures will respond to a change in the mass of firms in each industry. Beinga central supplier does not mean that entry or exit in your industry will have any effects on thesupplier centralities of other industries. Similarly, being a central consumer does not imply that
23
entry or exist in your industry will have any effects on the consumer centralities of other industries.To be influential, a firm must be central both as a consumer and as a supplier.
5.2.3 Equilibrium Impact of Shocks
Using lemmas 5.2 and 5.3 together, we can figure out the general equilibrium impact of a shock toan industry. For ease of notation, let
Λ =∂ log share of sales
∂ log(M)=∂ log(β) + log(α)
∂ log(M)= diag(β)−1Ψ1 + diag(α)−1Ψ2,
which is the partial-equilibrium elasticity of the sales distribution of industries with respect to themass of entrants M. Note that if k is an isolated industry, that is, an industry with connections onlyto the household, then Ψs
ik = Ψdki = 0 for i , k. Lemmas 5.2 and 5.3 then imply that Λik = Λki = 0
for all i , k. In other words, even though k can have very high supplier centrality β, consumercentrality α, and sales β × α, its importance to other industries through the extensive margin canbe zero. Interconnectedness matters directly in ways not captured by standard sufficient statistics.
To isolate the effect of the shocks through just the extensive margin, I consider shocks to thefixed costs fk rather than to labor productivity. Such shocks only propagate through the networkdue to the change in masses, and therefore give rise to cleaner analytical expressions. However,to keep the results comparable to standard models, I show the effect of labor productivity shocks(which have both intensive and extensive margin effects) on output in Appendix C.
Proposition 5.4. Let ek denote the kth standard basis vector. Then, in equilibrium, we have(d log(M1)d log( fk)
)= −(I −Λ)−1
(ek − 1
dPσc C/d fkPσc C
−
)= −
∞∑t=0
Λtek
︸ ︷︷ ︸network effect
+ constk︸︷︷︸general equilibrium
.
The interpretation of this expression is reminiscent of a shock in traditional input-outputmodels. A fixed cost shock to industry k affects the mass of firms in industry k, which in turncauses entry in upstream and downstream industries, which in turn causes even more entry, andso on ad infinitum. The term (I − Λ)−1 captures the cumulative effect. The crucial difference hereis that Λ is not the input-output matrix anymore. Once again, if an industry k is isolated, then allthe non-diagonal elements of the kth row and kth column of (I −Λ)−1 are zero.
We can also investigate how these fixed cost shocks to an industry affect aggregate output, orwelfare.8 To do see, it helps to define a new centrality measure.
8The results for labor productivity shocks are similar and can be found in appendix C.
24
Definition 5.1. The exit centrality v is given by
v ≡ β′︸︷︷︸tastes
Ψ2︸︷︷︸costs
(I −Λ)−1︸ ︷︷ ︸extensive margin
∝d log Cd log P
d log Pd log M
d log Md log f
Proposition 5.5. With free entry,d log(C)/d log( fk)d log(C)/d log( f j)
=vk
v j.
This vector v is the relevant notion of an industry’s centrality with respect to small changes tothe number of entrants in that industry. This measure has an intuitive interpretation. The effecton output, of a change in the mass of entrants, can be boiled down to the effect on the pricesof goods and services the household buys. The chain rule means that this can be broken downinto how consumption responds to price changes (tastes term), how prices respond to entry (pricesterm), and how entry responds to shocks (extensive margin term). Since we have independentlycharacterized each of these terms, it’s a matter of combining them. Observe that the exit centralitydoes not capture the effect on total sales, or the effect on total entrants, rather it translates theseeffects into its ultimate effect on the consumption of the household.
Unlike the centrality measures relevant in standard input-output models, like Acemoglu et al.(2012) and Baqaee (2015), this measure depends directly on interconnectedness, and on eachindustry’s role as both a supplier and as a consumer. Furthermore, it responds endogenously toshocks, meaning that the network structure is not summarized by a single vector.
5.2.4 Examples
The exit centrality vk of industry k loosely depends on how many weighted directed walks fromthe household to the factor of production (labor), and directed walks from the factor of production(labor) to the household, are severed by its removal. To make this more concrete, let us consider asimple supply chain depicted in figure 4. In this example, only industry 3 directly hires workers,and only industry 1 sells directly to the household. Industry 2 is a mediator who funnels paymentsfrom the household to labor and output from labor to the household.
In figure 5, I depict how the exit centrality v2 of industry 2 changes as a function of changes to themass of upstream and downstream firms. We see that industry 2 becomes more exit-central whenit has both many suppliers and many consumers; increasing only the number of 2’s consumersor only its suppliers does not affect v2 by nearly as much as increasing both at the same time,illustrating the fact that exit-centrality depends both on in-degrees and out-degrees.
In general, exit centralities can change for many different reasons. To isolate the effects of thenetwork structure on exit centrality, let us restrict our attention to examples where the networkis the only source of heterogeneity among firms. In other words, set β, α, ε,M to be uniform.
25
HH 1 2 3
wl
p1y1 p2y2 p3y3
Figure 4: Simple supply chain, where the solid arrows represent the flow of goods and the dashedarrows represent the flow of money. Only industry 3 directly employs labor and only industry 1sells to the household.
11.2
1.41.6
1.82
1
1.5
20
5
10
15
20
25
Mass of Consumers
Exit Centrality of Industry 2
Mass of Suppliers
Figure 5: Exit centrality v2 as a function of M1 and M3 for the economy depicted in figure 4. Forthis illustration, σ = 1.3, ε = 21,M2 = 1.
Next, consider the examples in figure 6. For each panel, we can work out which industry is mostinfluential on the extensive margin. The exit centralities of these examples are shown in figure 7.We see that for the simple supply-line in panel 6a, the middle industry is the most exit-central.On the other hand, for the production trident depicted in panel 6b, the intermediary industry, thatconnects three upstream industries with one downstream industry, is the most exit-central.
These examples illustrate that the more intuitively interconnected nodes, with many arrowspointing in and out, are the ones that have the highest exit centralities. Note that exit centralityneed not be related to sales. In fact, it is easy to write examples where the most exit-central playersdo not have the highest sales in the economy. For instance, consider the economy in figure 8. Theisolated industry has the highest sales, but industry 2 is more exit central.
26
1 2 3 4 5
(a) Production line
1 2
3
4
5(b) Production Trident
Figure 6: Two illustrative examples of how exit centrality can differ for different structures. For bothexamples, the household sector is omitted from the diagram. Let β = (1/5, 1/5, 1/5, 1/5, 1/5), σ =1.3,M = 1, ε = 31. Finally, let choose αk = 0.2
27
1 2 3 4 50
0.01
0.02
0.03
0.04
0.05Exit Centrality for Production Line
Industries
1 2 3 4 50
0.005
0.01
0.015
0.02
0.025Exit Centrality for Production Trident
Industries
Figure 7: Exit centralities for panel 6a and panel 6b. For this example, the only source of hetero-geneity is the network structure.
28
1 2
3
4
5
HH
1 2 3 4 50
0.05
0.1
0.15
0.2
0.25Exit Centrality
Industries1 2 3 4 5
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1Sales
Industries
Figure 8: An example where exit centrality is different to share of sales in a significant way.Industry 2 is the most exit-central, but industry 1 has the highest sales. Let β = (2/3, 1/3, 0, 0, 0), σ =1.3,M = 1, ε = 31, σ = 1.5. Finally, let choose αk = 0.2
29
5.3 Lumpy Firms
In this section, we consider the equilibrium where ‖∆‖ > 0. As argued by Gabaix (2011) andSutton and Trefler (2015), certain industries are dominated by very large firms. It turns out that thenonlinear propagation mechanism in this model can behave very differently when firms are notinfinitesimal. An equilibrium will now feature as many firms in each industry as possible, in thesense that adding a firm to any industry would drive that industry’s profits to be negative. Themodel with finitely many firms is a formidable combinatorial problem. Pure-strategy equilibrianeed not always exist. Technically, the non-existence of a pure-strategy equilibrium means that thismodel of the economy can feature non-fundamental or non-exogenous randomness. Intuitively,pure-strategy equilibria can fail to exist due to cycling, where the entry of a firm causes the profitsof another industry to go negative. Once a firm from that industry exists, another firm can enterthat causes the profits of the first entrant to be negative, and so on.
An equilibrium when ‖∆‖ > 0 is a solution to a nonlinear integer programming problem.Results from computational complexity theory suggest that we cannot hope to fully characterizethe set of equilibria. A naive brute-force computation would become infeasible very rapidly, sincewith even a few hundred firms, we may need to consider more cases than there are atoms in theobservable universe! Therefore, a full-fledged theoretical analysis of the model with lumpy firmsis left for future work. In this paper, I simply highlight the importance these granularities can haveby way of illustrative examples. Our approach here will be to compare the discontinuous model’sequilibria to the equilibria of the continuous model, with the hopes of gleaning some intuition. Todemonstrate some of the subtleties, consider the following example illustrated in figure 9.
6 5 4 3 2
1
HH.9 .9 .9 .9 2/5
3/5
Figure 9: Two production lines selling to the household HH. Industries 2 − 6 form on productionline and industry 1 forms a second (degenerate) production line. The direction of arrows representthe flow of goods and services. To keep the numerical example simple, I assume the Dixit-Stiglitzmarket structure.
30
β = (3/5, 2/5, 0, 0, 0, 0)′ ,
f = (0.01, 0.01, 0.01, 0.01, 0.01, f6),
σ = 1.1,
ε = (3, 3, 3, 3, 3, 1.2).∆ = (1, 1, 1, 1, 1, 0.5).
Let us consider the equilibrium mass of entrants M(∆, f6) for different values of f6 and mass vector∆. First, let us consider the massless limit,
M(0, 0.04) =
15.717.610.15.83.31.2
, M(0, 0.05) =
16.916.29.45.33.00.8
, M(0, 0.06) =
17.815.48.74.92.70.6
, M(0, 0.07) =
18.714.68.24.62.50.5
.
We see how increasing the fixed costs of the final supplier in the long chain reduces the mass offirms active in the long chain and increases the mass of firms in the competing short chain in acontinuous and intuitive way.
Now, let us consider this same equilibrium away from the massless limit.
M(∆, 0.04) =
161710531
, M(∆, 0.05) =
1915842
0.5
, M(∆, 0.06) =
1914842
0.5
, M(∆, 0.07) =
3300000
.
In going from f6 = 0.04 to f6 = 0.05, the discontinuous model amplifies the impact of the shockbecause the shock is large enough to force a discontinuous change. From f6 = 0.05 to f6 = 0.06however, the discontinuous model attenuates the impact of the shock since the firms are largeenough that they can absorb the losses without exiting. This represents a phase transition sinceshocks below a threshold are attenuated, and above that threshold, they are amplified. In goingfrom f6 = 0.06 to f6 = 0.07, the interior equilibrium disappears, and the long supply chain collapses.Furthermore, this is the unique equilibrium, so this is not a coordination failure resulting fromequilibrium switching. The situation is illustrated in figure 10.
This example suggests that under some conditions, we can use the continuous model to approx-imate the lumpy model. But it also warns that this approximation can break down dramatically.
31
1 2 3 40
5
10
15
20
25
30
Industries
Mas
ses
Mass of entrants for continuous model
1 2 3 40
5
10
15
20
25
30
Industries
Mas
ses
Mass of entrants for discrete model
123456
123456
Figure 10: Masses of industries with different levels of f6. Note that the discrete model amplifiesthe shock when going from the first panel to the second panel, attenuates it in going from thesecond to the third, and experiences a chain reaction in going from the third to the fourth.
32
1 2 3 4 5 60
0.05
0.1
0.15
0.2
0.25
Industries
Exit centrality
1 2 3 4 5 60
0.2
0.4
0.6
0.8
Industries
Sales
1 2 3 4 5 60
0.2
0.4
0.6
0.8
Industries
Supplier Centrality
1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
Industries
Consumer Centrality
Figure 11: Various measures of influence plotted for the mass of entrants M(∆, 0.04) as an example.We see that the exit centrality measure is not at all related to any of the other measures of centrality.Specifically, the final industry is the most exit-central, despite having the lowest overall sales.Recall that supplier centrality β measures the importance of the industry as a supplier to thehousehold, and the consumer centrality α measures the importance of the industry as a consumerof labor inputs from the household.
These differences are not only large, but they can also be very complex. Figure 11 illustrates thatwhen dealing with firm entry and exit, even in an example as simple as this one, the size, number,sales and prices of firms in an industry provide a poor guide to how important those industriesare to the equilibrium. As it happens, despite having the lowest share of sales, industry 6 has thehighest exit centrality. On the other hand, industry 1 is by far the largest industry, features thehighest sales, and the lowest costs, but it has the smallest exit centrality.
5.4 Approximation Error
A combinatorial analysis of the lumpy model is left for future work, however, we can gain someintuition for the behavior of this model by comparing its behavior to the limiting case ∆ → 0.We can ask when should we expect the continuous model, which is tractable, to provide a goodapproximation to the lumpy model, which is intractable. The lumpy model can behave verydifferently to the continuous model for reasons that are related to numerical instability in findingthe roots of certain systems of equations. In numerical analysis, such problems are called ill-conditioned. In this section, I show that a similar intuition applies to this problem, where thedifference between the lumpy and continuous model explodes under economically interpretable
33
conditions.To formalize the approximation error, we need a few definitions. These definitions have some
resemblance to the concepts of natural friends and natural enemies in international trade theory.Motivated by the Stolper and Samuelson (1941) result, Deardorff (2006) defines an industry to be anatural friend (enemy) of a factor when an increase in its prices will increase (decrease) the returnsto that factor. Inspired by these terms, I define the following.
Definition 5.2. Let πi(M1, . . . ,MN) be the profit function of industry i. Industry j is an enemy ofindustry i when
∂πi
∂M j< 0,
and industry j is a friend of industry i when
∂πi
∂M j> 0.
Industry j is a frenemy of industry i when∂πi
∂M j
takes both positive and negative values.
Although the definition is reminiscent of the one in Deardorff (2006), these definitions areout-of-equilibrium. In other words, we change the mass of firms in one industry without changingthe masses in other industries.
When industries are frenemies, the entry decisions of firms in one industry are strategic sub-stitutes for some regions and strategic complements for other regions. When industries are sub-stitutes, frenemies can only occur with non-degenerate (non-diagonal) input-output connections.
Proposition 5.6. Let σ > 1. When the network structure Ω is degenerate (diagonal), all industries areenemies. When the network is non-degenerate, industries can be frenemies.
It turns out that we can guarantee that approximating the discontinuous model with a contin-uous limit will be bad when firms are frenemies.
Theorem 5.7. Let M(∆) correspond to the mass of firms in each industry in an equilibrium where firm-levelmasses are given by ∆. Let M(0) denote equilibrium masses in each industry when ∆→ 0. Let
‖Dπ(M)‖ = sup ‖Dπ(M)‖ : M ∈ coM(0),M(∆) , ‖Dπ(M)‖ = sup‖Dπ(M)−1
‖ : M ∈ coM(0),M(∆).
As long as these exist, then
‖π(M(∆))‖‖Dπ(M)‖−1≤ ‖M(∆) −M(0)‖ ≤ ‖π(M(∆))‖‖Dπ(M)−1
‖.
34
In particular, the error gets larger when the profit functions have flat slopes, which can onlyoccur if industries are frenemies. In other words, the approximation errors explode when theentry decisions of firms in two industries switch from being strategic substitutes to being strategiccomplements. On the other hand, we see that when profits are small, then the discrete equilibriumwill be close to the continuous equilibrium. For the discrete model and continuous model to givesimilar answers, the profit function, and inverse profit function, must not change dramaticallywhen there are small changes in the mass of players. Or in the language of Trefethen and Bau(1997), these conditions ensure that we get “nearly right answers to nearly right questions.”
5.5 Bail-outs and the Nature of Externalities
The example in figure 9 suggests that bail-outs and government interventions may be desirable inthe lumpy model, since the failure of an important firm or industry can take down entire parts of anetwork, and cause a jump from one equilibrium to another.9 However, figuring out which failuresare efficient and which ones are not, in general, is very difficult. To implement the optimum for thediscrete model, a social planner would not only need access to an infeasible amount of information,but it would also have to solve an intractable computational and coordination problem.
Instead, inspired by the example where the president of Ford testified in favor of GM, we canimagine a scenario where the policy-maker can, in response to a shock, survey firms if other firmsshould be rescued. The firms will truthfully tell the policy-maker whether a rescue would increasetheir profits or decrease their profits. Is it the case that we can trust the testimony of other firms,particularly if those firms compete with the party that received the shock? Unfortunately, theanswer to this question is, in general, negative. To see this consider the example in figure 12, andsuppose that GM is hit with a negative fixed cost shock. If GM’s exit causes D to also exit, then itcan be in F’s interest to lobby for a bailout even if it is more efficient for the entire left-hand side ofthe figure to disappear.
To see this, recall that the profit function of a firm in industry k is given by
g(Nk)Pσc Cβk × αk − fk,
where g is a decreasing function of the number of competitors Nk in industry k. Denote theequilibrium with no bail-out as NB and the equilibrium with a bail-out as B. Ideally, we wouldwant to find a firm i such that
πBi > π
NBi ⇒ CB > CNB.
However, the presence of the network terms β, α, the price-level term Pσc , and the competition
9In fact, even in the continuous limit, as long as the network-structure is not degenerate, the entry margin is inefficientin this model. Loosely speaking, double-marginalization means that upstream industries, appropriately defined, receivetoo little demand and are too small in equilibrium. A full analysis of optimal industrial policy is beyond the scope ofthis paper.
35
GM F
HH
D
M TM
B
Figure 12: A stylized example of a supply chain. HH is household; GM is General Motors; F is FordMotor Company; D is Dunlop Tires; M is Mitsubishi Motors; T is Toyota; and, B is BridgestoneTires, a Japanese tire manufacturer. The arrows represent the flow of goods and services.
term g(Nk) mean that profits may not move in the same direction as real GDP. So, F may wantGM rescued to protect D, even if it results in lower consumption C. In general, it is impossibleto guarantee that another firm’s testimony is reliable, even if that firm is completely disconnectedfrom origin of the shock. So that, in figure 12, even pure competitors like T or M may not givereliable feedback on whether or not GM should be bailed out.
6 Conclusion
This paper highlights the importance of firm entry and exit in propagating and amplifying shocksin a production network. I show that without the extensive margin, canonical input-outputmodels have several crucial limitations. First, a firm’s influence depends solely on the firm’s roleas a supplier to other firms, and its role as a consumer is irrelevant. Second, each firm’s influencemeasure is exogenous, and the exogenous influence measures are sufficient statistics for the input-output matrix. In this sense, for every input-output model, there exists a non-interconnected modelwith different parameters that has the same equilibrium responses. Third, a firm’s influence iswell-approximated by the firm’s size, so the nature of interconnections does not matter as long asit results in the same distribution of firm sizes. These limitations imply that, in these models, if anindustry is systemically important, then it must be big, and that if an industry is big, then it mustbe systemically important.
However, when the mass of firms in each industry can adjust endogenously, all of theseproperties disappear. The influence of an industry is endogenous, depends on its role as a supplierand as a consumer of inputs, and is not well approximated by its size. Two industries with thesame demand-chains can have very different effects on the equilibrium if their supply-chains aredifferent, and vice versa. Furthermore, there are no sufficient statistics that summarize the network.
36
In this sense, the model is not isomorphic to a non-network model with different parameters.Despite these features, I show that in the limit where all firms have zero mass, the model with
entry is very analytically tractable. I derive a new measure of systemic importance called exitcentrality that ranks industries by how an exit or entry in that industry would affect output. Ishow that exit centrality, while intuitive to interpret, is unrelated to that industry’s sales or itsprices. An important next step, left for future work, is to estimate these exit centralities in the data.
I also show that when firms in some industries have positive mass, or are “granular,” thena failure of one of these systemically important firms can snowball into an avalanche of failures.Such domino chain reactions are impossible in standard input-output models. I show that we canexpect these cascades to occur when firms’ entry decisions switch from being strategic substitutesto being strategic complements. A more thoroughgoing analysis of the model’s properties in thepresence of granularities is another avenue for future work.
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A Appendix: No Extensive-Margin and Monopolistic Competition
In section 4, we assume that there are no fixed costs of entry and all industries are perfectlycompetitive. In this subsection, I consider the case where there is no entry but firms have somemarket power and positive fixed costs. This makes it easier to understand how adding theextensive margin will affect the results, since the extensive margin will only matter once we havefixed costs and product differentiation. I state a few key propositions in this subsection to showhow the intuition of the previous section will carry over to this case.
First, let us consider the special case where the elasticity of substitution across industries σ = 1,which is the Cobb-Douglas case.
Proposition A.1 (Productivity shock). Let the elasticity of substitution across industries be equal to one,then
log(C(z|E)) = const + β′(I −Ω)−1(α log(z)).
That is, the network-structure Ω, which has N2 parameters, is summarized by N sufficient statistics. Thesesufficient statistics are β′(I −Ω)−1 = β′Ψd.
The sufficient statistic β′Ψd is closely related to the sales shares β = β′Ψs. To see this, note that
β = β′∞∑
t=0
(µ−1Ω
)t,
where µ is the diagonal matrix of mark-ups. Whereas,
β′Ψd = β′∞∑
t=0
(Ω)t .
Therefore, the relevant statistics β′Ψs are the sales shares that would prevail if there were nomark-ups. Intuitively, they are still an exogenous supplier-centrality measure.
Proposition A.2 (Productivity shock). When the elasticity of substitution across industries is not equalto one and there is no entry or exit, then
C(α|E) =1 −M′ f
1 − diag(ε)−1β′α/β′α
(β′α
) 1σ−1 ,
= f (β′ΨsΨdα, β′Ψdα) = f (β′Ψdα, β
′Ψdα).
That is, the network-structure Ω, which has N2 parameters, is summarized by 2N sufficient statistics: β′Ψd
and β′Ψd.
β is just the supplier centrality we have dealt with previously. It is an exogenous suppliercentrality, and depends solely on the amount of household demand that reaches the industry,
40
whether directly through retail sales, or indirectly through other industries. As before, the exactnature of the network-structure is irrelevant since an industry could be big because it sells a lot tothe household (large β) or because it supplies many other firms (large Ψdei).
The Cobb-Douglas case constitutes a very special knife-edge scenario where, because expen-diture shares are exogenous, the length of supply chains do not matter. Once we allow for marketpower εk < ∞, and endogenous expenditure shares σ , 1, each time funds flow from one industryto another, they are attenuated by monopoly profits. Longer chains accrue larger mark-ups, andthis changes the expenditure shares. Due to this effect, we need a measure that not only takesthe intensity of supply chains into account, but also their length. This is the role that the secondsufficient statistic β′ΨsΨd = β′Ψd plays. This measure is related to the upstreamness measuredefined by Antras et al. (2012), since it not only takes into account the strength of the connectionsbut also their length.
To see this, suppose that we have a unit mass of firms in each industry M = I and that allmark-ups are zero. In this extreme case, Ψd = Ψs. So, we can interpret
ΨdΨs = (I −Ω)−2 = I + 2Ω + 3Ω2 + 4Ω3 + . . . .
This calculation gives some intuition for why β′ΨsΨd is a supply-side centrality measure thatcontrols for the length of the supplier relationship as well as its intensity.
Proof of proposition A.1. By theorem 3.5,
∑k
πk =β′
εαPcC − 1′M f ,
where division by ε is elementwise. Observe that
PcC = (1 +∑
k
πk).
Therefore, ∑k
πk =
(β′
εα − 1′M f
)1
1 − β′
ε α.
Therefore, nominal GDP
PcC = 1 +
(β′
εα − 1′M f
)1
1 − β′
ε α,
does not respond to shocks. Therefore, real GDP is given by
log(C) = const − log(Pc).
41
Sincelog(Pc) = −β′(I −Ω)−1 (
α log(z))
+ const,
we can writelog(C) = const + β′(I −Ω)−1 (
α log(z)).
Proof of proposition A.2. ∑k
πk =β′
εαPσc C − 1′M f ,
where division by ε is elementwise. Observe that
Pσc C = PcCPσ−1 =(1 +
∑k πk)
β′α.
Combine these two expressions to get the desired result.
Local Amplification
The following proposition shows that the equilibrium effect of a productivity shock zi to industryi’s sales, in percentage change terms, is greatest for industry i. Furthermore, the effect on otherindustries decays geometrically according to the sum of geometric matrix series Ψs. Since salesare proportional to gross profits, the proposition also applies to gross profits.
Proposition A.3. For any economy E, if we fix the mass of firms in each industry, we have
d log(salesi)dzi
−d log(sales j)
dzi=
Ψsiiαi
αi−
Ψsjiαi
α j≥ 0.
Proof of proposition A.3. Using the same argument as in the proof of proposition D.1, we can showthat
d log(sales j)
dzσ−1i
=1α j
e′jΨαiei +1
Pσc CdPσc Cdzσ−1
i
,
This means thatd log(salesi)
dzσ−1i
−d log(sales j)
dzσ−1i
= αi
e′iΨei
α j−
e′jΨei
α j
,as required. Lemma D.2 then implies that this is always greater than zero.
42
B Appendix: Proofs
Lemma B.1. Demand for the output of firm i in industry k is
y(k, i) = c(k, i) +∑
l
Nl∑j
x(l, j, k, i)∆l,
= βk
(p(k, i)
pk
)−εk
M−ϕkεkk
(pk
pc
)−σC +
∑l
Mlωlk
(p(k, i)
pk
)−εk
M−ϕkεkk
(pk
λl
)−σy(l, j),
where λl is the marginal cost of the representative firm in industry l.
Proof. Cost minimization by each firm implies firm j in industry l’s demand for inputs from firm iin industry k is given by
x(l, j, k, i) = ωlk
(p(k, i)
pk
)−εk
M−ϕkεk
(pk
λl
)−σy(l, j),
where λl is the marginal cost of firms in industry l,
λk =
αkzσ−1k w1−σ +
∑l
ωklp1−σl
1
1−σ
,
and pk is the price index for industry k
pk =
M−ϕkεkk
Nk∑i
∆kp(k, i)1−εk
1
1−εk
.
Household demand for goods from firm i in industry k are
c(k, i) = βk
(p(k, i)
pk
)−εk
M−ϕkεkk
(pk
Pc
)−σC,
where Pc is the consumer price index
Pc =
∑k
βkp1−σk
1
1−σ
.
Adding the household and firms’ demands together gives the result, assuming the symmetricequilibrium.
43
Proof of lemma 3.1. By lemma B.1
y(k, i) = βkM−ϕkεkk p(k, i)−εkpεk−σ
k pσc C +∑
l
MlωlkM−ϕkεkk p(k, i)−εkpεk−σ
k λσl y(l, j).
In equilibrium, we can substitute pk = M1−ϕkεk
1−εkk p(k, i) to get
M1−ϕkεkεk−1 εk−ϕkεkpσk y(k, i) = βkpσc C +
∑l
Mlλσl ωlky(l, j). (2)
Observe that, in equilibrium,
yk =
M−ϕkk
Nk∑∆ky(k, i)
εk−1εk
εkεk−1
= M1−ϕkεk−1 εk
k y(k, i).
Furthermore1 − ϕkεk
εk − 1ε + ϕkεk =
1 − ϕk
1 − εkεk.
Therefore, we can rewrite (2) as
pσyk = βkPσc C +∑
l
Mlωlkλσl y(l, j).
Now substitute in
λl = µ−1l p(l, i) = µ−1
l plM1−ϕlεlεl−1
l
to get
pσyk = βkPσc C +∑
ωlkM1+
1−ϕlεlεl−1 σ−
1−ϕlεlεl
εl
l µ−σl pσl yl.
Finally, note that
1 +1 − ϕlεl
εl − 1σ −
1 − ϕlεl
εlεl =
σ − 1εl − 1
(1 − ϕlεl).
Recall that M is defined to be the diagonal matrix whose kth diagonal element is M1−ϕkεkεk−1
k , and µ isthe diagonal matrix whose kth element is industry k’s markup. So, denote sk = pσk yk. This meansthat we can write
s′ = β′Pσc C + s′Mσ−1µ−σΩ.
44
Rewrite this to get
s′ = β′(I − Mσ−1µ−σΩ)−1Pσc C
= β′Pσc C.
Proof of proposition 3.2. Let A = µ−σM1−σΩ. Note that, by assumption, A jk = A jl for every j. LetA(n)
ji denote the jith entry of An. We wish to show, by induction, that
A(n)jk = A(n)
jl , (n ∈ 1, 2, . . .).
To that end, fix n ∈ N and assume that
A(n−1)jk = A(n−1)
jl .
Then, it follows that
A(n)jk =
∑i
A jiA(n−1)ik ,
=∑
i
A jiA(n−1)il ,
= A(n)jl .
Since the induction assumption holds for n = 1 by assumption, it follows that
A(n)jk = A(n)
jl , (n ∈ 1, 2, . . .).
Finally, observe that
βk =
∞∑n=0
β jA(n)jk ,
= βk +
∞∑n=1
β jA(n)jk ,
which we have shown is equal to
= βk +
∞∑n=1
β jA(n)jl ,
45
which, since βk = βl, equals
=
∞∑n=0
β jA(n)jl ,
= βl.
Proof of proposition 3.3. Similar to the proof of proposition 3.2.
Proof of lemma 3.4. Since all firms have constant returns to scale on the margin, firm i’s problem inindustry k, conditional on entry, can be written as By definition
p(k, i) = µkλk.
Substituting this into the definition of λk we get
p(k, i) = µk
αkzσ−1k w1−σ +
∑l
ωklp1−σl
1
1−σ
.
Note that, in the symmetric equilibrium,
pk = µkM1−ϕkεk
1−εkk λk,
so,
µσ−1k M
1−ϕkεkεk−1 (1−σ)
k p1−σk = αkzσ−1
k w1−σ +∑
l
ωklp1−σl .
Let P be the vector pk and let P1−σ represent elementwise exponentiation. Then
µσ−1M1−σP1−σ = (α zσ−1)w + ΩP.
Rearrange this to get
P1−σ = (I − µ1−σMσ−1Ω)−1µ1−σMσ−1(α zσ−1
)w1−σ.
Proof of theorem 3.5. Note that the profits of firm i in industry k are
π(k, i) = p(k, i)y(k, i) − λky(k, i) − w fk.
46
This is equivalent to
π(k, i) = p(k, i)y(k, i) −1µk
p(k, i)y(k, i) − w fk,
=µk − 1µk
p(k, i)y(k, i) − w fk.
Since all active firms in industry k are identical this is
π(k, i) =µk − 1µk
1Mk
pkyk − w fk.
By lemmas 3.1 and 3.4,pkyk = pσk ykp1−σ = βkαkPσc Cw1−σ,
and soπ(k, i) =
µk − 1µk
1Mk
βkαkPσc Cw1−σ− w fk.
Proof of proposition 4.2. Note that industry k’s sales are given by
pkyk = pσk ykp1−σ = βkαkPσc Cw1−σ.
Observe that in equilibrium with no shocks,
α = (I −Ω)−1α = 1.
Therefore,pkyk = βkPσc Cw1−σ.
So an industry’s shares of sales relative to other industries is determined solely by βk.
Proof of theorem 4.1. Note that in this special case Ψs = Ψd = (I − Ω)−1. To simplify notation, letΨ = (I − Ω)−1. First, let us consider real GDP C(z|E) as a function of productivity shocks z for aCobb-Douglas economy E.
Lemma B.2 (Productivity shock). Let the elasticity of substitution across industries be equal to one, then
log(C(z|E)) = β′(α log(z)) =
N∑k=1
wlkGDP
log(zk).
That is, the network-structure Ω which has N2 parameters is summarized by N sufficient statistics. Thesesufficient statistics are each industry’s expenditures on labor as a share of GDP.
47
This lemma is a slight generalization of results in Acemoglu et al. (2012). If we assume thateach industry’s labor share is constant, so that αi = α for all i, and household expenditure sharesare uniform so that βi = 1/N, then we exactly recover the result in Acemoglu et al. (2012), whichtells us that the effect of a productivity shock on real GDP depends on share of sales.
Proof of proposition B.2. Note that real GDP can be written as
C =PcCPc
=wl + π
Pc,
where π is total profits. By free entry, profits are zero in equilibrium. Normalize w = 1. Then
log(C) = − log(Pc).
The marginal costs of firms in industry k are given by
λk =(zk
w
)−αk ∏l
pωkll .
Since industries are perfectly competitive, firms set prices equal to their marginal costs. Let Pdenote the vector of industry prices. Then, in equilibrium,
log(P) = (I −Ω)−1 (α (log(w) − log(z))
).
Therefore,
log(C) = − log(Pc),
= −β′ log(P),
= −β′(I −Ω)−1 (−α log(z)
),
= β′α log(z).
Note that β is sales as a share of GDP, and βkαk is therefore industry k’s wage bill as a share ofGDP.
Corollary (Productivity shock). Let the elasticity of substitution across industries be equal to one, andall industries have the same labor share α, then
log(C(z|E)) = αN∑
k=1
pkyk
GDPlog(zk).
That is, the network-structure Ω which has N2 parameters is summarized by N sufficient statistics. These
48
sufficient statistics are each industry’s share of sales.
These propositions imply that when σ = 1, the supplier centrality β is a sufficient statisticfor translating productivity shocks into real GDP. In particular, the aggregate impact of a vectorof shocks depends on the supplier centrality weighted average of the productivity shocks. Aslemma 3.1 shows, βk is equal to industry k’s share of sales. Therefore, the bigger the industryin equilibrium, the larger the impact of its productivity shocks on GDP. The exact nature of thenetwork-structure is irrelevant since an industry i could be big because it sells a lot to the household(large βi) or because it supplies many other firms (the ith column of Ψ is large).
These intuitions also carry over to the case where σ , 1.
Lemma B.3 (Productivity shock). When the elasticity of substitution across industries is not equal toone, then
C(α|E) =(β′α
) 1σ−1 .
That is, the network-structure Ω, which has N2 parameters, is summarized by N sufficient statistics: β.
Outside of the Cobb-Douglas case, the supply-side centrality β′ = β′Ψs is no longer an indus-try’s share of sales. Lemma 3.1 shows that it is still related to an industry’s share of sales however.In fact, even though β′ is not equal to share of sales globally, it is still closely related to share ofsales.
Real consumption is given bywl + π
Pc=
1Pc.
We have shown thatPc =
(β′(I −Ω)−1α
) 11−σ .
Let β′ = β′(I −Ω)−1 to get the desired result.
Proof of proposition 5.1. Under condition (a), the markups at the industry level are exogenous anddetermined by the industry-level elasticity of substitution µk = εk/(εk − 1). To derive markupsunder condition (b), note that firm i in industry k face the following demand function
y(i, k) =
(p(i, k)
pk
)−εk
yk,
=
(p(i, k)
pk
)−εkβk
(pk
Pc
)−σC +
∑l
ωlkMl
(pk
λl
)−σy(l, j)
,Under condition (b), the firm internalizes its effects on the industry-level aggregates pk and yk (butnot on economy-wide aggregates like Pc and C) when it sets its prices. Then we can write the profit
49
function of firm i in industry k as being proportional to
p(i, k)1−εkpε−σk − λip(i, k)−εpε−σk .
Maximizing this gives the first order condition
(1 − εk)p(i, k)−εkpεk−σk + (εk − σ)p(i, k)1−εkpεk−σ−1
k ∆k
(p(i, k)
pk
)−εk
+
λip(i, k)−1−εkpεk−σk − λp(i, k)−εk(εk − σ)pεk−σ−1
k ∆k
(p(i, k)
pk
)−εk
= 0.
Rearrange this to get
(1 − εk) + (εk − σ)∆k
(p(i, k)
pk
)1−εk
+ λiεk
p(i, k)− λi(εk − σ)∆k
(p(i, k)
pk
)1−εk
= 0.
In the symmetric equilibrium, this is
(1 − εk) +εk − σ
Nk+λiεk
p(i, k)− λi
εk − σNk
1p(i, k)
= 0.
Rearrange this to get the optimal price as
p(i, k) = µkλi =εkNk − (εk − σ)
(εk − 1)Nk − (εk − σ)λi.
In the limit, as Nk →∞, this gives an exogenous markup of εk/(εk − 1). In the case where the firmis a monopolist, and Nk = 1, this gives the markup σ/(σ − 1). To make this sensible, we need thatεk > σ > 1.
Finally, to derive markups under condition (c), let εk → ∞, so that all firms in industry k areproducing the same product. Now note that demand for industry k’s output is given by
yk =
Nk∑j
y(k, j) = p−σk const,
where const depends on economy-wide variables the firm treats as exogenous. The firm choosesits quantity to maximize profits
pky(k, i) − λkyk,
which gives the first order condition
const1σ − const
1σ
(1σ
) y(k, i)yk
= λky1σ
k .
50
Rearrange this to get
pk = λk
(1 −
1σNk
)−1
.
Proof of lemma 5.2. Recall thatβ′ = β′(I − SΩ)−1.
So
dβ′
dMi= −β′Ψs
d(I − SΩ)dMi
Ψs,
= β′ΨsdS
dMiS−1SΩΨs,
= β′Ψsd log S
dMi(Ψs − I),
= β′d log S
dMi(Ψs − I).
Sodβk
dMi= βi
d log Si
dMi
(Ψs
ik − 1(i = k)).
Assuming the Dixit-Stiglitz structure, so that S = Mσ−1µ−σ means,
β′ = β′(I − Mσ−1µ−σΩ)−1.
So
dβ′
d log(Mi)= Mi
dβ′
dMi,
= −Miβ′Ψs
d(I − Mσ−1µ−σΩ)dMi
Ψs,
= Miβ′Ψs
dMσ−1
dMiµ−σΩΨs,
= Miβ′Ψs
dMσ−1
dMiMσ−1M1−σµ−σΩΨs,
= Miβ′Ψs
dMσ−1
dMiMσ−1(Ψs − I),
51
The kth element of this vector is
dβk
d log(Mi)=σ − 1εi − 1
βi(Ψs − I)ek.
Putting this all into a matrix gives
dβ′
d log(M1)= diag(β)diag
(σ − 1ε − 1
)(Ψs − I).
Proof of lemma 5.3. Recall thatα = (I −DΩ)−1Dα.
So
dαdMi
= (I −DΩ)−1 dDdMi
Ω(I −DΩ)−1Dα + (I −DΩ)−1 dDdMi
α,
= (I −DΩ)−1DD−1 dDdMi
D−1DΩ(I −DΩ)−1Dα + (I −DΩ)−1 dDdMi
α,
= ΨdD−1 dDdMi
D−1(ΨdD−1I)Dα + ΨdD−1 dDdMi
α,
= Ψdd log D
dMiD−1α.
Sodαk
dMi= αi
d log Di
dMiΨd
ki1
Di.
Assuming the Dixit-Stiglitz structure, so that D = Mσ−1µ−σ means,
α = (I − Mσ−1µ1−σΩ)−1Mσ−1µ1−σα.
To simplify the notation, for this proof, let
B = (I − Mσ−1µ1−σΩ)−1.
52
So
1Mi
dαd log(Mi)
=dα
dMi,
= BdMσ−1
dMiµ1−σα + B
d(Mσ−1)dMi
(Mσ−1
)−1(Mσ−1)µ1−σΩB(Mσ−1)µ1−σα,
= BdMσ−1
dMiµ1−σα + B
d(Mσ−1)dMi
(Mσ−1
)−1(B − I)(Mσ−1)µ1−σα,
= Bd(Mσ−1)
dMi
(Mσ−1
)−1B(Mσ−1)µ1−σα,
= Bd(Mσ−1)
dMi
(Mσ−1
)−1α,
= Ψd
(M
)1−σµσ−1 d(Mσ−1)
dMi
(Mσ−1
)−1α,
The kth element of this vector is
dαk
d log(Mi)=
(σ − 1εi − 1
) (εi
εi − 1
)σ−1e′kΨdeiαi
1Mσ
i.
Putting this all into a matrix gives
dαdM
= Ψddiag(α)µσ−1diag(M)1−σdiag(σ − 1ε − 1
).
Proof of proposition 5.4. Observe that, with the Dixit-Stiglitz structure,
log(M1) = log(β) + log(α) + log(Pσc C)1 − log( f ) − log(ε).
Hence,d log(M1)d log( fi)
= Λd log(M1)d log( fi)
+d log(Pσc C)
d log( fi)1 − ei.
Rearrange this to getd log(M1)d log( fi)
= (I −Λ)−1(
d log(Pσc C)d log( fi)
1 − ei
).
Proof of proposition 5.5. Recall thatlog(C) =
(β′α
) 1σ−1 ,
53
whenced log(C)d log( fi)
=1
σ − 11β′α
β′dα)
d log(M1)d log(M1)d log( fi)
.
Sinced log(Pσc C)
d log( fi)= −(σ − 1)
d log(C)d log( fi)
,
We can write
d log(C)d log( fi)
=1
σ − 11β′α
β′Ψ2(I −Λ)−1(
d log(Pσc C)d log( fi)
1 − ei
),
= −1
σ − 11β′α
β′Ψ2(I −Λ)−1(ei + (1 − σ)
d log(C)d log( fi)
1),
= −1
1 + 1β′αβ
′Ψ2(I −Λ)−11σ − 1β′α
β′Ψ2(I −Λ)−1ei.
Divide d log(C)/d log( fi) by d log(C)/d log( f j) to get the result.
Proof of proposition 5.6. By theorem 3.5, the profits of industry i are
πi =βiαi
fiεiPσc C,
when Ω = 0. Thereforedπi
dM j=βiαi
fiεi
dPσc CdM j
.
We can writePσc C =
1 −M′ fβ′α − diag(ε)−1β′α
.
Hence,
dPσc CdM j
=− f j
β′α − diag(ε)−1β′α−
1 −M′ fβ′α − diag(ε)−1β′α
βi(1 − 1/εi)σ − 1εi − 1
Mσ−εiεi−1
i µσ−1i αi.
Since σ > 1, both of these terms are negative and industries can only be enemies.
Proof of theorem 5.7. Let π(M) be the vector industrial profits with mass of entrants M. Note thatan equilibrium of the continuous limit corresponds to M(0) such that π(M(0)) = 0. Let M(∆)correspond to the equilibrium mass of entrants for some ∆ 0. Then π(M(∆)) ≥ 0. By themean-value inequality
‖π(M(∆))‖ = ‖π(M(∆)) − π(M(0))‖ ≤ ‖Dπ(M∗)‖‖M(∆) −M(0)‖,
54
where M∗ is in the convex hull of M(0) and M(∆). Rearrange this expression to get the left handside of the inequality. A similar computation on π−1 gives the right hand side, as long as π−1 existsover coM(0),M(∆)
C Appendix: Labor Productivity Shocks and the Extensive Margin
Proposition C.1. The derivative of the equilibrium mass of firms in each industry M with respect to a laborproductivity shock in industry k is given by(
d log(M)dzk
)= (I −Λ)−1
(1dPσc C/dzk
Pσc C+ diag(α)−1Ψdek
).
This expression is very intuitive. We can rewrite the expression in proposition C.1 as(d log(M)
dzk
)=
∞∑t=0
Λt(
1dPσc C/dzk
Pσc C+ diag(α)−1Ψsek
)=
∞∑t=0
Λt(
1dPσc C/dzk
Pσc C
)︸ ︷︷ ︸
GE effect
+
∞∑t=0
Λt(diag(α)−1Ψsek
)︸ ︷︷ ︸
network-structure effect
.
First, consider the “network-structure” term. The initial impact of a productivity shock to k,holding fixed the mass of firms in each industry, is given by diag(α)−1Ψsek – this is the traditionalinput-output effect which captures the total intensity with which each industry uses inputs fromindustry k. In proposition A.3, I show this term captures the extent to which industry sales respondto shocks when there is no entry/exit. However, when we have entry/exit, the traditional input-output effect must be multiplied by a new term (I − Λ)−1. This captures how the industry’s sizeschange in response to the intensive margin shock. Recall that
Λ =∂ log(β) + log(α)
∂ log(M)∝∂ log(share of sales)
∂ log(M),
where the proportionality sign follows from lemmas 3.1 and 3.4. Therefore, (I −Λ)−1 captures thecumulative effect of the shock on the size of the industries.
So we see that in equilibrium, a productivity shock to industry k will first affect the masses inall other industries through its effect on the marginal costs of anyone who buys from k. However,the initial change in masses causes the supplier and consumer centralities of all industries tochange (captured by Ψ1 and Ψ2). This change in centrality measures, in turn, causes the massesto adjust again, and this changes the centralities again, and so on ad infinitum. This gives riseto a new amplification channel captured by the geometric series of Λ. The cumulative effect on
55
the equilibrium of these changes is captured by the “network-structure effect” term. The shockalso has an effect on the general price level and real GDP, and the second “GE effect” captures thisgeneral equilibrium effect. It is a simple matter to show that in equilibrium, dPσc C/dzk is just aweighted average of the network-structure effects.
Proof of proposition C.1. Note that
Mi =βiαi
εi fiPσc C.
Therefore,
d log(M)dαi
= M1dPσc C/dαi
Pσc C+ Mdiag(β)−1 dβ
dMdMdαi
+ Mdiag(α)−1 dαdM
dMdαi
+ Mdiag(α)−1Ψdei.
Rewrite this as
d log(M)dαi
= M1dPσc C/dαi
Pσc C+ M
(diag(β)−1Ψ1 + diag(α)−1Ψ2
) dMdαi
+ Mdiag(α)−1Ψdei.
Rearrange this to get the desired result.
Now, let us analyze how real GDP responds to productivity shocks in this massless limit.This result should be compared to proposition A.2, which gave the response when there was noextensive margin of entry and exit. Finally, it helps to compare the output responses of this modelto the canonical input-output model.
Proposition C.2. With free entry,
d log(C)/d log(αk)d log(C)/d log(α j)
=v(diag(α)−1 + I
)Ψdek
v(diag(α)−1 + I
)Ψde j
.
The presence of exit centrality v, which depends on Ψ1 and Ψ2, and therefore depends on in-degrees and out-degrees, shows that the most influential industry is not simply the industry whosupplies the most or consumes the most. It also provides an additional channel of amplification tothe purely intensive margin result. Furthermore, output is no longer linearly related to productivityshocks, due to the nonlinear propagation mechanisms of entry and exit. Lastly, since v and Ψd
are now endogenous, and depend on how many firms are in each industry, the sufficient statisticsapproach no longer works.
Proof of proposition C.2. Note that
C =PcCPc
=1 +
∑k πk
Pc=
(1
β′Ψd(α zσ−1)
) 11−σ
.
56
Therefore,
d log(C)dzi
=1
σ − 1
d(log
(β′Ψd(α zσ−1)
))dzi
,
=1
σ − 11β′α
β′∂α∂M
∂M∂zi
+ β′Ψdd(α zσ−1)
dzi,
by lemma 5.3 and proposition C.1,
=1
σ − 11β′α
β′Ψ2MM−1(I −Λ)−1(1
dPσc C/dzi
Pσc C+ diag(α)−1Ψdei
)+
1β′α
β′Ψdeiαizσ−2i .
Note thatdPσc C/dzi
Pσc C=
d log(Pσc C)dzi
= −(σ − 1)d log(C)
dzi.
Substituting this into the previous expression and rearranging gives(1 +
β′Ψ2(I −Λ)−11β′α
)d log(C)
dzi=
1β′α
(β′Ψ2(I −Λ)−1
(diag(α)−1 + I
)Ψdei
)αizσ−2
i .
Rearranging this gives the desired result.
D Appendix: Local Amplification
Let us briefly investigate the nature of shock propagation in the version of the model withoutentry. The following proposition shows that the equilibrium effect of a productivity shock zi toindustry i’s sales, in percentage change terms, is greatest for industry i. Furthermore, the effect onother industries decays geometrically according to the sum of geometric matrix series Ψ.
Proposition D.1. The semi-elasticity of firm i’s sales relative to firm j’s sales with respect to a shock to firmi’s productivity around the steady state z = 1 is given by
d log(salesi)
dzσ−1i
−d log(sales j)
dzσ−1i
= αi
(Ψii −Ψ ji
)> 0.
Proof of proposition D.1. To show this, first we need the following technical lemma.
Lemma D.2. Let A be a non-negative N ×N matrix whose rows sum to one. Let a, b, c be N × 1 vectors in
57
the unit cube with c = a + b. Let B = (I − diag(1 − c)A)−1. Then
e′i Bei
e′i Ba≥
e′jBei
e′jBa(i , j),
where ei is the ith standard basis vector. When a is strictly positive, then the inequality is strict.
The sales of industry j are given by
β je′jΨ(α zσ−1
)Pσc C.
Therefore
d log(sales j)
dzσ−1i
=1β j
dβ j
dzσ−1i
+1α j
de′jΨ(α zσ−1
)dzσ−1
i
+1
Pσc CdPσc Cdzσ−1
i
,
= 0 +1α j
de′jΨ(α zσ−1
)dzσ−1
i
+1
Pσc CdPσc Cdzσ−1
i
.
Note thatdΨ
(α zσ−1
)dzσ−1
i
∣∣∣∣∣∣∣z=1
= Ψd(α zσ−1
)dzσ−1
i
∣∣∣∣∣∣∣z=1
= Ψαiei.
Substitute this in to getd log(sales j)
dzσ−1i
= e′jΨαiei +1
Pσc CdPσc Cdzσ−1
i
,
where we use the fact that α = 1. This means that
d log(salesi)
dzσ−1i
−d log(sales j)
dzσ−1i
= αi
(e′iΨei − e′jΨei
),
as required. Lemma D.2 then implies that this is always greater than zero.
This proposition formalizes the intuition that a shock has to decay as it travels, and the input-output connections Ω control the rate of decay. The rate of decay can be slow enough to overturnthe law of large numbers, as shown by Acemoglu et al. (2012). However, even though the networkcan slow the decay, proposition D.1 shows that shocks cannot be amplified as they travel outfrom their source. In order for shocks to industry k to have a big impact, industry k must havestrong connections to the household, or strong connections to other industries that have strongconnections to the household, and so on. But having strong connections to the household impliesthat the industry has to be large in equilibrium. The lack of local amplification means that sizeand influence are closely linked in this class of models. This, and the earlier results in this section,
58
show that as long as firms are small, and they do not supply large fractions of the economy, thenshocks to them cannot have significant aggregate effects.
Local Amplification with Entry
The equilibrium impact of a change in an industry’s productivity at the firm level is very stark.
Proposition D.3. In equilibrium, the effect of a productivity shock zk on the sales of firm i in industry k isthe same as its effect on firm j in industry l.
d log(saleski)dzk
−d log(salesl j)
dzk= 0.
At the firm level, the mass of firms in each industry adjusts to ensure that all firms are equallyexposed to productivity shocks. At the industry-level, the conclusion is less stark.
Proof of proposition D.3. Let s be the sales of the representative firm in each industry. Then
d log(s)dα
=ddα
(log(β) + log(α) − log(M)
)+
ddα
log(Pσc C).
By the chain rule,
d log(s)dα
=
(diag(β)−1 dβ
dM+ diag(α)−1 dα
dM− diag(M)−1
)dMdα
+ diag(α)−1Ψdek +ddα
log(Pσc C),
=(diag(β)−1Ψ′1diag(M)−1 + diag(α)−1Ψ2diag(M)−1
− diag(M)−1) dM
dα
+ diag(α)−1Ψdek +ddα
log(Pσc C),
=(diag(β)−1Ψ′1 + diag(α)−1Ψ2 − I
)diag(M)−1 dM
dα+ diag(α)−1Ψdek +
ddα
log(Pσc C),
=[(
diag(β)−1Ψ′1 + diag(α)−1Ψ2 − I)
diag(M)−1diag(M)(I − diag(β)−1Ψ′1 − diag(α)−1Ψ2
)−1+ I
]diag(α)−1Ψdek +
ddα
log(Pσc C),
= 0,
where the second to last line uses proposition C.1.
Proposition D.4. In equilibrium, the difference in the response of the profits of industry k relative toindustry j to a productivity shock to industry k is given by
d log(sk)dzk
−d log(s j)
dzk= (ek − e j)′(I −Λ)−1
(diag(α)−1Ψsek
). (3)
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Proof of proposition D.4. Let s be the sales of the industries. Then
d log(s)dα
=ddα
(log(β) + log(α) + log(Pσc C)
).
By the chain rule,
d log(s)dα
=
(diag(β)−1 dβ
dM+ diag(α)−1 dα
dM
)dMdα
+ diag(α)−1Ψdek +ddα
log(Pσc C),
=(diag(β)−1Ψ′1diag(M)−1 + diag(α)−1Ψ2diag(M)−1
) dMdα
+ diag(α)−1Ψdek +ddα
log(Pσc C),
=[Λdiag(M)−1diag(M)(I −Λ)−1 + I
] [diag(α)−1Ψdek +
ddα
log(Pσc C)],
= (I −Λ)−1[diag(α)−1Ψdek +
ddα
log(Pσc C)],
which gives the desired result. Note that the second to last line of the derivation uses propositionC.1, and the last line uses the fact that
Λ(I −Λ)−1 + I = (I −Λ)−1.
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